Are there any interesting features of transcendental numbers, other than the fact that they are hard to write down?

Maybe they don't seem special to me because I never really cared about fractions, and the details of "Fill in the gaps" to get real numbers was just taxonomy rather than exploration to me?IE:Start with positive numbers.Add zero. (sketchy as a step IMO, but an interesting story)Extend to negatives.Fill in the gaps between the integers.Extend into complex numbers.

SuicideJunkie wrote:Are there any interesting features of transcendental numbers, other than the fact that they are hard to write down?

No. Unless you're a mathematician.

SuicideJunkie wrote:Maybe they don't seem special to me because I never really cared about fractions, and the details of "Fill in the gaps" to get real numbers was just taxonomy rather than exploration to me?IE:Start with positive numbers.Add zero. (sketchy as a step IMO, but an interesting story)Extend to negatives.Fill in the gaps between the integers.Extend into complex numbers.

It was just taxonomy. For all practical purposes, rational numbers are fine, because any real number can be approximated to any desired degree of precision by rational numbers.

It's easy to dismiss the constructivist approach to mathematics from where we are, but it was a natural thing to try, even if it turned out to be a dead end (just like the search for the luminiferous ether turned out to be).

The first sign of trouble with the constructivist approach was the ancients' failure to solve the problem of squaring the circle. That problem was proven to be impossible when pi was proven to be transcendental, more than 2000 years after people had first started studying it.

Pfhorrest wrote:I was under the impression that people still defended constructivism today. Are they just crazy or are there legitimate arguments still ongoing?

I couldn't say about number theory. Maybe someone will come up with a scheme that can construct all the transcendental numbers? Although I suspect that the non-enumerability of R should make that impossible.

In the broader sense, Gödel's Incompleteness Theorem put constructivism in mathematics to rest, but that was about constructing all valid mathematical theorems, not about constructing or enumerating numbers. But I should confess at this point that I'm not a mathematician myself (although I did study math in college) so the finer points of this are over my head.

SuicideJunkie wrote:Are there any interesting features of transcendental numbers, other than the fact that they are hard to write down?

That's a pretty good summary of it. Their most interesting feature is that almost every number is transcendental, but very few of the numbers we use are. In fact, the vast majority of numbers are not only transcendental, but also impossible to specify with any finite description. Which gets weird if you think about it...

cellocgw wrote:Irrationals are theoretical, since we can't write them out in any finite decimal notation.

You can write down an algorithm to create pi. But you can't write down the decimal representation of ⅑.

Integers are theoretical: you can't show me "two." You can show me two objects, and you can make a symbol that represents "two" (Hint: try "2" ), but "two" is just a concept. [feel free to use that argument to troll any meta-math discussion you like]

Theoretical concepts do exist, while the objects you'd show are just sensory input.